Dice Expectation and Variance

Question

A standard fair six-sided die is thrown $1800$ times, where outcomes in different throws are stochastically independent. Let $X$ be the number of times the die shows either $2$ or $3$. Name the distribution which $X$ follows. Give the corresponding weights, $E[X]$ and $Var(X)$.

My attempt:

Distribution: X follows the binomial distribution with $X $~$ bin(1800,\frac{1}{3})$

Weights: For integers $k$ in the interval $0 \leq k \leq 1800$, $P(X=k) = 1800Ck \cdot \frac{1}{3}^k \frac{2}{3}^{(1800-k)}$, and $0$ otherwise.

$E[X] = np = 1800 \cdot \frac{1}{3} = 600$

$Var(X) = np(1-p) = 600 \cdot \frac{2}{3} = 400$

Are my answers correct ?

Answer 1

Yes indeed! Your working is correct. Good job.